Geometrical Quantities
The components of curvature tensors such as the Einstein tensor have, in geometric units, the dimensions of sectional curvature. So do the components of the stress-energy tensor. Therefore the Einstein field equation is dimensionally consistent in these units.
Path curvature is the reciprocal of the magnitude of the curvature vector of a curve, so in geometric units it has the dimension of inverse length. Path curvature measures the rate at which a nongeodesic curve bends in spacetime, and if we interpret a timelike curve as the world line of some observer, then its path curvature can be interpreted as the magnitude of the acceleration experienced by that observer. Physical quantities which can be identified with path curvature include the components of the electromagnetic field tensor.
Any velocity can be interpreted as the slope of a curve; in geometric units, slopes are evidently dimensionless ratios. Physical quantities which can be identified with dimensionless ratios include the components of the electromagnetic potential four-vector and the electromagnetic current four-vector.
Physical quantities such as mass and electric charge which can be identified with the magnitude of a timelike vector have the geometric dimension of length. Physical quantities such as angular momentum which can be identified with the magnitude of a bivector have the geometric dimension of area.
Here is a table collecting some important physical quantities according to their dimensions in geometrized units. They are listed together with the appropriate conversion factor for SI units.
| Quantity | SI dimension | Geometrical dimension | Multiplication factor |
|---|---|---|---|
| Length | 1 | ||
| Time | c | ||
| Mass | G c −2 | ||
| Velocity | 1 | c −1 | |
| Angular velocity | c −1 | ||
| Acceleration | c −2 | ||
| Energy | G c −4 | ||
| Energy density | G c −4 | ||
| Angular momentum | G c −3 | ||
| Force | 1 | G c −4 | |
| Power | 1 | G c −5 | |
| Pressure | G c −4 | ||
| Density | G c −2 | ||
| Electric charge | G 1/2 c −2 (4πε0) −1/2 | ||
| Electric potential | 1 | G 1/2 c −2 (4πε0) 1/2 | |
| Electric field | G 1/2 c −2 (4πε0) 1/2 | ||
| Magnetic field | G 1/2 c −1 (4πε0) 1/2 | ||
| Potential | 1 | G 1/2 c −1 (4πε0) 1/2 |
This table can be augmented to include temperature, as indicated above, as well as further derived physical quantities such as various moments.
Read more about this topic: Geometrized Unit System
Famous quotes containing the word quantities:
“The Walrus and the Carpenter
Were walking close at hand:
They wept like anything to see
Such quantities of sand:
“If this were only cleared away,”
They said, “it would be grand!”
“If seven maids with seven mops
Swept it for half a year,
Do you suppose,” the Walrus said,
“That they could get it clear?”
“I doubt it,” said the Carpenter,
And shed a bitter tear.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)